A generative model based on the 2/3 power law

crazy_paths.jl

using Random

function crazy_paths(N,delta_t)

	ddx, dx = 10*(2*rand(N+1) .-1.0), 10*(2*rand(N+1) .-1.0)
	ddy, dy = 10*(2*rand(N+1) .-1.0), 10*(2*rand(N+1) .-1.0)

	x, y = 10*(2*rand(N+1) .-1.0), 10*(2*rand(N+1) .-1.0)

	K = abs(ddx[1]*dy[1]-dx[1]*ddy[1])

	A, B = zeros(N), zeros(N)

	for i = 1:N

		## the transformation:
		alpha, beta, theta = (-1)^rand(1:2)*rand(0.1:0.01:10), (-1)^rand(1:2)*rand(0.1:0.01:10), rand()
		delta = 2*rand(2) .-1.0 ## a translation vector which doesn't change the volume

		M = [cos(2*pi*theta)/alpha sin(2*pi*theta)*beta; -1*sin(2*pi*theta)/beta cos(2*pi*theta)*alpha];

		A[i], B[i] = M[1], M[3]

		## update the variables: 
		#ddx[i+1] = M[1]*ddx[i] + M[3]*dx[i] .+ delta[1]
		#ddy[i+1] = M[1]*ddy[i] + M[3]*dy[i] .+ delta[1]

		#dx[i+1] = M[2]*ddx[i] + M[4]*dx[i] .+ delta[2]
		#dy[i+1] = M[2]*ddy[i] + M[4]*dy[i] .+ delta[2]

		## update the variables: 
		ddx[i+1] = M[1]*ddx[i] + M[3]*dx[i]
		ddy[i+1] = M[1]*ddy[i] + M[3]*dy[i] 

		dx[i+1] = M[2]*ddx[i] + M[4]*dx[i] 
		dy[i+1] = M[2]*ddy[i] + M[4]*dy[i] 

		## update:
		x[i+1] = x[i] + dx[i]*delta_t + 0.5*ddx[i]*delta_t^2
		y[i+1] = y[i] + dy[i]*delta_t + 0.5*ddy[i]*delta_t^2

	end

	q_4 = ddy[1]*A[1] + dy[1]*B[1]
	q_2 = ddx[1]*A[1] + dx[1]*B[1]

	return x,y,q_4, q_2

end


function test_hypothesis(N,delta_t)

	output = zeros(100)

	for i=1:100

		x,y,q_4, q_2 = crazy_paths(N,delta_t)

		range = rand(20:80)

		lower, upper = rand(1:N-range), rand(N-range+1:N)

		expected_slope = q_4/q_2

		actual_slope = (y[upper]-y[lower])/(x[upper]-x[lower])

		max_slope = max(expected_slope,actual_slope)
		min_slope = min(expected_slope,actual_slope)

		sign_exp, sign_act = abs(expected_slope)/expected_slope, abs(actual_slope)/actual_slope

		if sign_exp*sign_act == 1.0 && max_slope/min_slope < 2.0

			output[i] = 1.0

		end
	end

	return output


end